Non-linear QCD evolution with improved triple-pomeron vertices

In a previous publication, we have constructed a set of non-linear evolution equations for dipole scattering amplitudes in QCD at high energy, which extends the Balitsky–JIMWLK hierarchy by including the effects of fluctuations in the gluon number. In doing so, we have relied on the color dipole picture, and we have neglected the non-locality of the two-gluon exchanges in the relation between scattering amplitudes and dipole densities. In this Letter, we relax the latter approximation, and thus restore the correct structure of the ‘triple-pomeron vertex’ which describes the splitting of one BFKL pomeron into two in the terms responsible for fluctuations. The ensuing triple-pomeron vertex coincides with the one previously derived by Braun and Vacca within perturbative QCD. The evolution equations can be recast in a Langevin form, but with a multivariable noise term with off-diagonal correlations. Our equations are shown to be equivalent with the modified version of the JIMWLK equation recently proposed by Mueller, Shoshi, and Wong.     

A QCD analysis of diffractive deep-inelastic scattering data

We perform a novel type of analysis of diffractive deep-inelastic scattering data, in which the input parton distributions of the pomeron are parameterised using the perturbative QCD expressions. In particular, we treat individually the components of the pomeron of different size. We are able to describe simultaneously both the recent ZEUS and H1 diffractive data. In addition to the usual two-gluon model for the perturbative pomeron, we allow for the possibility that it may be made from two sea quarks.Received: 12 July 2004, Revised: 3 September 2004, Published online: 24 September 2004     

Diffractive parton distributions from perturbative QCD

The asymptotic collinear factorisation theorem, which holds for diffractive deep-inelastic scattering, has important modifications in the sub-asymptotic HERA regime. We use perturbative QCD to quantify these modifications. The diffractive parton distributions are shown to satisfy an inhomogeneous evolution equation. We emphasise that it is necessary to include both the gluonic and sea-quark t-channel components of the perturbative pomeron. The corresponding pomeron-to-parton splitting functions are derived in the appendix.Received: 20 April 2005, Published online: 27 July 2005     

Magnitude of Regge cut contributions in the triple-Regge region

Starting from the reggeon calculus, the various possibilities of absorptive pomeron cut corrections in the triple-Regge region are considered. For the case of pp → pX, we estimate their importance at present day energies. We conclude that at highest ISR energies, pomeron cuts of the eikonal type are not enough, and enhanced diagrams with at least one additional triple-pomeron coupling need be included.     

Pomeron structure function and diffraction dissociation of virtual photons in perturbative QCD

We study scaling properties of the diffraction dissociation of virtual photons in a deep inelastic scattering. We concentrate on the total diffraction dissociation rate, diffraction excitation mass spectrum and the pomeron structure function to the lowest order in perturbative QCD. We calculate the valence structure function and the strangeness and charm content of the pomeron and estimate the ocean structure function using the pomeron factorization property. We find that quarks carry ≈ 10% of pomeron's momentum. Differential cross section of the (virtual) photon-pomeron scattering is found to exhibit features typical of the hadronic two-body reactions, supporting a treatment of the Pomeron as a particle, whereas the flavor dependence of structure functions does not support the particle treatment of the pomeron. Diffraction dissociation of photons is predicted to make ≈ 15% of the total deep inelastic scattering rate at smallx and largeQ ². Detailed predictions for the mass spectrum and angular distribution of jets produced on the valence component of the pomeron are presented.     

The triple pomeron interaction in the perturbative QCD

The triple pomeron interaction is studied in the perturbative approach of BFKL-Bartels. At finite momentum transfers √?t the contribution factorizes in the standard manner with a triple-pomeron vertex proportional to 1/√?t, Att=0 the contribution is finite, although it grows faster with energy than for finitet and does not factorize.     

Pomeron loops in the perturbative QCD with large <Emphasis Type="Italic">N</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>

The lowest order pomeron loop is calculated for the leading conformal weight with full dependence of the triple-pomeron vertex on intermediate conformal weights. The loop is found to be convergent. Its contribution to the pomeron Green function begins to dominate already at rapidities 10–15. The pomeron pole renormalization is found to be quite small due to a rapid fall of the triple-pomeron vertex with rising conformal weights.     

BFKL pomeron propagator in the external field of the nucleus

It is shown by numerical calculations that the convoluted QCD pomeron propagator in the external field created by a solution of the Balitsky-Kovchegov equation in the nuclear matter vanishes at high rapidities. This may open a possibility to apply the perturbative approach for the calculation of pomeron loops.     

The relation between the Good–Walker and triple-regge formalisms for diffractive excitation

In this Letter we analyze the relation between the triple-pomeron and Good–Walker formalisms for diffractive excitation in DIS and hadronic collisions. In both approaches gap events are interpreted as the shadow of absorption into inelastic channels. We here argue that the two formalisms are just different views of the same phenomenon. We first demonstrate how this relation works in a simple toy model, and then show how the relevant features of the toy model are also realized in real perturbative QCD.     

Pomeron vertices in perturbative QCD in diffractive scattering

We analyze the momentum space triple pomeron vertex in perturbative QCD. In addition to the standard form of this vertex which is used in the context of total cross sections at high energies and in the QCD reggeon field theory, there exists an alternative form which has to be used in the study of high-mass diffraction. We review and analyze the relation between these two versions. We discuss some implications for the BK equation. In the second part of our paper we extend this analysis to the pomeron-odderon-odderon vertex.Received: 11 January 2005, Revised: 24 January 2005, Published online: 3 March 2005     

On the d<Subscript>T</Subscript>=0 toy model in reggeon field theory

Reggeon field theory with zero transverse dimensions is critically reanalyzed in the Hamiltonian formulation for both a sub- and a supercritical pomeron. Different mathematical aspects of the model, starting from the scalar products in the space of quantum states, are discussed. The probabilistic picture is addressed in the absence of pomeron merging. The issue of the large loop approximation is discussed in terms of the Hamiltonian evolution and its relation to the probabilistic picture recalled. A perturbative treatment, based on the PT symmetry of the model, is proposed that may be useful for more realistic models. Finally, we present numerical calculations for the various parameters of the models, α(0)-1=μ and the triple-pomeron coupling constant λ, which help one to understand some mathematical aspects and the different approximation regimes. They show that the triple-pomeron interaction always makes amplitudes fall with rapidity, irrespective of the value of the intercept. The smaller the values of the ratio λ/μ, the higher are the rapidities y at which this fall starts, so that at small values of λ it begins at asymptotically high rapidities (for λ/μ<1/4 the fall is noticeable only at μy>100). No visible singularity is seen for the critical pomeron.     

Energy flow and energy correlations in deep inelastic scattering

We study two examples of infrared-safe quantities in deep inelastic scattering: the flow of energy in a given angular range and the energy-energy angular pattern. We derive expressions for these quantities in perturbative QCD, to order α_s, and show explicitly their infrared safety. Our formulas for the angular energy flow and the energy-energy angular pattern depend only on well-defined QCD factors and on the deep inelastic structure functions. To gauge whether or not these perturbative QCD results are applicable to present day data, we estimate in a simple model the effects of hadronization and primordial parton transverse momentum. In general these non-perturbative effects mask the resulting QCD pattern at present energies, but vanish more rapidly at higher energies than the QCD effects. However, we point out two examples where it may be possible to test the perturbative QCD predictions with available data. One of them involves studying the x-dependence of the azimuthal asymmetry of the energy flow. The other involves studying the angular width of the energy-energy correlation function.     

A momentum space analysis of the triple pomeron vertex in pQCD

We study the properties of the momentum space triple pomeron vertex in perturbative QCD. Particular attention is given to the collinear limit where the transverse momenta on one side of the vertex are much larger than on the other side. We also comment on the kernels in the non-linear evolution equations.     

Scaling properties of nuclear shadowing in deep inelastic scattering

We present the perturbative QCD analysis of nuclear shadowing in deep inelastic scattering and prove it to be the scaling, rather than the higher twist, 1/Q², effect, contrary to numerous recent claims. We argue in favor of the scaling triple-pomeron contribution to the nuclear shadowing and comment on the mass spectrum of diffraction excitation by highly virtual photons. Our numerical predictions for shadowing are in good agreement with the recent EMC data.     

Conformal couplings and “azimuthal matching” of QCD pomerons

Using the asymptotic conformal invariance of perturbative QCD we derive the expression of the coupling of external states to all conformal spin p components of the forward elastic amplitude. Using the wave function formalism for structure functions at small x, we derive the perturbative coupling of the virtual photon for , which is maximal for linear transverse polarization. The non-perturbative coupling to the proton is discussed in terms of “azimuthal matching” between the proton color dipoles and the configurations of the photon. As an application, the recent conjecture of a second QCD pomeron related to the conformal spin-1 component is shown to rely upon a strong azimuthal matching of the component in –proton scattering. Received: 25 October 1999 / Revised version: 19 January 2000 / Published online: 14 April 2000     

Predictions for high-energy real and virtual photon–photon scattering from color dipole BFKL–Regge factorization

High-energy virtual photon–virtual photon scattering can be viewed as an interaction of small size color dipoles from the beam and target photons, which makes scattering at high energies (LEP, LEP200 and NLC) an indispensable probe of the short distance properties of the QCD pomeron exchange. Based on the color dipole representation, we investigate the consequences for the scattering of the incorporation of asymptotic freedom into the BFKL equation which makes the QCD pomeron a series of isolated poles in the angular momentum plane. The emerging color dipole BFKL–Regge factorization allows us to relate in a model-independent way the contributions of each BFKL pole to scattering and DIS off protons. Numerical predictions based on our early works on the color dipole BFKL phenomenology of DIS on protons are in good agreement with the experimental data on the photon structure function and the most recent data on the cross section from the OPAL and L3 experiments at LEP200. We discuss the role of non-perturbative dynamics and predict a pronounced effect of the Regge-factorization breaking due to large unfactorizable non-perturbative corrections to the perturbative vacuum exchange. We comment on the salient features of the BFKL–Regge expansion for scattering including the issue of the decoupling of subleading BFKL poles and the soft plus rightmost hard BFKL pole dominance. Received: 9 January 2001 / Revised version: 25 September 2001 / Published online: 7 December 2001     

An exact solution to reggeon calculus

By choosing a suitable but unconventional bare pomeron propagator and bare triple-pomeron coupling the Dyson equations of reggeon calculus are shown to reduce to algebraic equations for the various coupling constants. The resulting pomeron, which is identical to the input pomeron is the familiar black or gray disk.     

Shadowing effects in nuclear parton distributions for small values ofx

The shadowing corrections to gluon and quark distributions in nuclei in the region of small values ofx are discussed. They are related to parton distributions in a pomeron which are in principle measurable in hard diffractive processes on the nucleon target. Multiple scattering corrections to shadowing are considered in a model dependent way. The perturbative QCD evolution of shadowing is also taken into account. Various possibilities of the partonic content of a pomeron are considered. It is shown in particular that the conventional parametrizations of parton distributions in a pomeron which are based on the assumption that it consists mostly of gluons imply substantial nuclear shadowing in gluon distributions in heavy nuclei. Possible phenomenological implications of shadowing corrections in nuclear parton distributions for various semi-hard processes with nuclear targets are briefly discussed.     

Existence of a phase transition in reggeon field theory in two transverse dimensions

We show that a reggeon field theory, in the physical number of dimensions, with a linear bare trajectory and a constant triple-pomeron coupling, undergoes a phase transition as the bare pomeron intercept is increased through a certain critical value.     

Unitarity and the BFKL pomeron

High energy onium-onium scattering is calculated as a function of impact parameter in the one- and two-pomeron exchange approximation. Difficulties with using the multiple scattering series to unitarize single pomeron exchange at high energy are noted. An operator formalism which sums all numbers of pomeron exchange is given. A toy model which has a similar operator structure at high energy as QCD is presented and the S-matrix is evaluated. Estimates of the energies and impact parameters at which blackness occurs in onium-onium scattering are given. It is emphasized that the problem of unitarity in high energy onium-onium scattering can be solved in a purely perturbative context, with a non-running coupling if the onium is heavy enough.